Mechanics 1
Pearson Edexcel · International A-Level · Further Mathematics Topic 4 5:08 English narration · English + 中文 subtitles burned in
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Paper WME01: 1 hour 30 minutes, 75 marks, with P1 and P2 assumed.
WME01 试卷：1 小时 30 分钟，75 分，默认已学 P1 和 P2。
Take g as 9.8, give inexact answers to 2 or 3 significant figures, and memorise these: momentum and impulse, the suvat formulae, F equals m a with friction mu R, and the magnitude of a vector.
g 取 9.8，非精确答案保留 2 或 3 位有效数字，并记住这些：动量和冲量、suvat 公式、F = ma 和摩擦力 μR，以及向量的模。
The words in a stem are instructions.
题干中的词就是指令。
A particle has mass but no size.
质点有质量但没有大小。
Light means massless, so tension is unchanged.
轻表示无质量，所以张力不变。
Uniform puts the centre of mass at the midpoint.
均匀表示质心在中点。
Inextensible keeps both ends at the same speed.
不可伸长使两端速度相同。
Smooth means no friction.
光滑表示没有摩擦。
State the assumption you use.
要写明你用到的假设。
With i east and j north, a bearing is measured clockwise from north.
以 i 为东、j 为北，方位角从北方顺时针量。
The vector minus 5i minus 12j points south-west, 22.6 degrees west of due south, so its bearing is 180 plus 23: 203 degrees.
向量 −5i − 12j 指向西南，在正南偏西 22.6°，所以方位角是 180 + 23，即 203°。
A moves from minus 5i plus 16j to 10i plus 4j in 3 seconds, so its velocity is 5i minus 4j.
A 在 3 秒内从 −5i + 16j 移到 10i + 4j，所以速度是 5i − 4j。
Its position is 5t minus 15 i plus 24 minus 4t j.
位置是 (5t − 15)i + (24 − 4t)j。
At t equals 5, B minus A is minus 5i minus 12j: bearing 203 degrees.
t = 5 时，B − A = −5i − 12j：方位角 203°。
A ball is projected up from A, 12 metres above the ground, and lands after 4 seconds.
球从离地 12 米的 A 点竖直上抛，4 秒后落地。
Take up as positive from A: s is minus 12, so minus 12 equals 4U minus 4.9 times 16, and U is 16.6.
以 A 为原点、向上为正：s = −12，所以 −12 = 4U − 4.9 × 16，U = 16.6。
Then v squared is 510.8: it lands at 22.6 metres per second.
然后 v² = 510.8：落地速度 22.6 米每秒。
On a velocity-time graph the gradient is acceleration and the area is displacement.
在速度–时间图上，斜率是加速度，面积是位移。
Speed cannot go negative, so the throw is a V: falling from 16.6 to zero at the top, after 1.7 seconds, then rising to 22.6.
速率不能为负，所以这次抛掷是一个 V 形：在 1.7 秒时从 16.6 降到最高点的零，然后升到 22.6。
Both legs have gradient g.
两段斜率都是 g。
Two ropes pull a barge straight along the canal: 800 newtons at 25 degrees above the line of motion, 500 newtons below it.
两根绳子拉着驳船沿运河直线前进：800 牛顿在运动方向上方 25°，500 牛顿在下方。
Across the canal there is no acceleration, so the two sideways components must cancel: alpha is 43 degrees.
横跨运河方向没有加速度，所以两个侧向分量必须抵消：α = 43°。
Along the canal, the barge accelerates at 0.023.
沿运河方向，驳船的加速度是 0.023。
P, of mass k M, and Q, of mass 3M, hang over a smooth pulley and accelerate at g over 5.
质量为 kM 的 P 和质量为 3M 的 Q 挂在光滑滑轮两侧，加速度为 g/5。
Write one equation for each body: T minus k M g for P, 3 M g minus T for Q.
每个物体各写一个方程：P 是 T − kMg，Q 是 3Mg − T。
Adding eliminates T: k is 2.
相加消去 T：k = 2。
The string pulls the pulley down with 2T, 23.5 newtons.
绳子以 2T 向下拉滑轮，即 23.5 牛顿。
P, mass k m at 5u, catches Q, mass 3m at 2u; afterwards P moves at 2u.
质量 km、速度 5u 的 P 追上质量 3m、速度 2u 的 Q；之后 P 以 2u 运动。
The impulse on Q is 4.5 m u, so v is 3.5u.
Q 受到的冲量是 4.5mu，所以 v = 3.5u。
The impulse on P is equal and opposite: k m times 3u equals 4.5 m u, so k is 1.5.
P 受到的冲量大小相等、方向相反：km × 3u = 4.5mu，所以 k = 1.5。
On a rough surface the contact force splits into the normal reaction R and friction F.
在粗糙表面上，接触力分为法向反力 R 和摩擦力 F。
Moving, F equals mu R.
运动时 F = μR。
At rest, F is at most mu R, and only the phrase on the point of moving licenses equality.
静止时 F ≤ μR，只有“即将运动”才允许取等号。
With tan theta three quarters and mu a quarter: R is four fifths m g.
tan θ = 3/4，μ = 1/4：R = (4/5)mg。
Sliding up, weight and friction both act down the slope, so the deceleration is four fifths g, kept exact.
向上滑动时，重力分量和摩擦力都沿斜面向下，所以减速度是 (4/5)g，保持精确。
Held at rest by X up the plane, with mass 2, X is four fifths g: 7.84 newtons.
质量为 2 时，用沿斜面向上的力 X 使其静止，X = (4/5)g，即 7.84 牛顿。
A particle with weight 50 newtons hangs from two strings at 40 and 60 degrees to the vertical.
重 50 牛顿的质点挂在两根与竖直方向成 40° 和 60° 的绳子上。
Horizontally T one sine 40 equals T two sine 60; vertically the upward parts make 50.
水平方向 T₁ sin 40° = T₂ sin 60°；竖直方向向上的分量之和为 50。
So T one is 44 and T two is 33.
所以 T₁ = 44，T₂ = 33。
A rod AB of length 3a rests on the ledge at C, 2a from A.
长 3a 的杆 AB 搁在边缘 C 上，C 距 A 为 2a。
An upward P at B tilts it about A: 3a P equals W d.
B 处向上的力 P 使它绕 A 翻转：3aP = Wd。
A downward 1.25P at B tilts it about C: 1.25 P a equals W times 2a minus d.
B 处向下的力 1.25P 使它绕 C 翻转：1.25Pa = W(2a − d)。
Eliminating P, d is 24a over 17.
消去 P，d = 24a/17。
Check yourself.
自我检测。
A light string has the same tension throughout.
轻绳各处张力相同。
The area under a velocity-time graph is displacement.
速度–时间图下的面积是位移。
A smooth pulley feels 2T.
光滑滑轮受力 2T。
At rest, F equals mu R only on the point of moving.
静止时只有即将运动才有 F = μR。
And take moments about the point that removes unknowns.
取矩时选能消去未知量的点。